Question

Which of the following are eigen functions of the operator d2/dx2 , find the corresponding eigen value?

(i) Sin x                        (ii)Sin2x                           (iii) e2x​

Answer 1

All three of them are the eigen functions for the given operator.

Explaination,

Those functions which when acted upon by the operator gives the answer the function itself multiplied by a scalar, then the function is termed as eigen function of the given operator and the scalar is called eigen value.

Operator = $\frac{d^{2} }{dx^{2} }$

Operator(Function) = λFunction

$i)$ $Sinx$

Double differentiation of $Sinx$ gives

$\frac{d^{2}Sinx }{dx^{2} } = -Sinx$

Therefore, the eigen value is $-1$

$ii)$ $Sin2x$

Double differentiation of $Sin2x$ gives

$\frac{d^{2}Sin2x }{dx^{2} } = -4Sinx$

Therefore, the eigen value is $-4$

$iii)$ $e^{2x}$

Double differentiation of $e^{2x}$ gives

$\frac{d^{2}x^{2}e^{2x} }{dx^{2} } = 4e^{2x}$

Therefore, the eigen value is $4$.